Can a neural network be used to predict the next pseudo random number?
If we are talking about a perfect RNG, the answer is a clear no. It is impossible to predict a truly random number, otherwise it wouldn't be truly random.

When we talk about pseudo RNG, things change a little. Depending on the quality of the PRNG, the problem ranges from easy to almost impossible. A very weak PRNG like the one XKCD published could of course be easily predicted by a neural network with little training. But in the real world things look different.

The neural network could be trained to find certain patterns in the history of random numbers generated by a PRNG to predict the next bit. The stronger the PRNG gets, the more input neurons are required, assuming you are using one neuron for each bit of prior randomness generated by the PRNG. The less predictable the PRNG gets, the more data will be required to find some kind of pattern. For strong PRNGs this is not feasable.

On a positive note, it is helpful that you can generate an arbitrary amount of training patterns for the neural network, assuming that you have control over the PRNG and can produce as many random numbers as you want.

Because modern PRNGs are a key component for cryptography, extensive research has been conducted to verify that they are "random enough" to withstand such prediction attacks. Therefore I am pretty sure that it is not possible with currently available computational resources to build a neural network to successfully attack a PRNG that's considered secure for cryptography.

It is also worth noting that it is not necessary to exactly predict the output of a PRNG to break cryptography - it might be enough to predict the next bit with a certainty of a little more than 50% to weaken an implementation significantly. So if you are able to build a neural network that predicts the next bit of a PRNG (considered secure for cryptography) with a 55% success rate, you'll probably make the security news headlines for quite a while.